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Simulating the Forward Curve in the Two-Factor Hull–White Model

Article Quant Q&A · Author: Eddie E.
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Summary

The question describes a two-factor Hull–White short-rate model in which the short rate combines a deterministic shift, calibrated to the initial forward curve, with two mean-reverting Gaussian factors. It asks how to update the entire forward curve through time, rather than simulating only the short rate. The model allows the factors to have imperfectly correlated shocks, adding flexibility to the rate dynamics.

The response points to the Heath–Jarrow–Morton representation as a way to express the evolution of the forward curve, while cautioning that this representation does not make simulation simpler. It recommends using the model’s closed-form zero-coupon bond pricing formulas and deriving forward rates from those prices. The exchange supplies no derivation, implementation details, or numerical comparison, so it gives a modeling direction rather than a step-by-step simulation method.

Key ideas

  • The two-factor model represents the short rate as a deterministic shift plus two mean-reverting factors.
  • The deterministic shift is chosen to fit the initial forward curve.
  • Correlated factor shocks allow richer rate movements than a single-factor setup.
  • The Heath–Jarrow–Morton representation describes forward-curve evolution but may not simplify computation.
  • Closed-form zero-coupon bond prices are presented as a practical route to obtaining forward rates.

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Fixed incomeDerivatives pricingStatistics
Full text
# How does the 2-factor Hull White model propagate the forward rates curve?


# How does the 2-factor Hull White model propagate the forward rates curve?












I've been trying to get a grasp on some of the basics of interest rate modeling, and am looking to simulate rates using the 2 factor Hull White model, which I am aware offers a more realistic model of rates which allows for imperfect correlation between instantaneous forward rates.

I've found resources on the web which reduce the model to an additive Gaussian one, where one has for the short rate $r(t)$:

$r(t) = \varphi(t) + x_1(t) + x_2(t)$

where $x_1, x_2$ are mean reverting processes governed by:

$dx_1(t) = -a_1x_1(t)\cdot dx + \sigma_1\cdot dW_1(t)$

$dx_2(t) = -a_1x_2(t)\cdot dx + \sigma_2\cdot dW_2(t)$

with $dW_1(t)dW_2(t) = \rho$, and $\varphi(t)$ is deterministic and chosen to fit the initial forward rate curve $f(0,t)$:

$\varphi(t) = f(0,t) + \frac{\sigma_1^2}{2a_1^2}\left(1-e^{-a_1t}\right)^2+\frac{\sigma_2^2}{2a_2^2}\left(1-e^{-a_2t}\right)^2+\rho\frac{\sigma_1\sigma_2}{a_1a_2}\left(1-e^{-a_1t}\right)\left(1-e^{-a_2t}\right)$

This tells me how to simulate the short rate (by updating $x_1,x_2$ at each time increment and adding to $\varphi$), but my question is, how could one simulate the evolution of the whole forward curve? I have also found (unwieldy) closed-form expressions for $P(t,T)$ the price of a term $T$ zero coupon bond at time $t$, from which you can obtain the forward curve, but is there a way to generate the forward curve at time $t+\Delta t$ by updating the curve at time $t$, akin to the way we can do it for the short rate $r(t)$?

## Answer by experquisite (score 2, accepted)

https://quant.stackexchange.com/a/9110

The Heath-Jarrow-Morton representations of short interest rate models (such as Hull-White) will give you an expression for the evolution of the entire forward curve, but it doesn't make the problem any easier. The closed form ZC formulae you mention above are probably your best bet.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.